$12^{2}_{78}$ - Minimal pinning sets
Pinning sets for 12^2_78 Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^2_78 Pinning data
Pinning number of this multiloop: 4
Total number of pinning sets: 640
of which optimal: 5
of which minimal: 5
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 3.12357
on average over minimal pinning sets: 2.6
on average over optimal pinning sets: 2.6
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
• {1, 4, 6, 10}
4
[2, 2, 3, 3]
2.50
B (optimal)
• {1, 4, 9, 10}
4
[2, 2, 3, 3]
2.50
C (optimal)
• {1, 4, 8, 10}
4
[2, 2, 3, 4]
2.75
D (optimal)
• {1, 4, 9, 11}
4
[2, 2, 3, 4]
2.75
E (optimal)
• {1, 3, 4, 9}
4
[2, 2, 3, 3]
2.50
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
4
5
0
0
2.6
5
0
0
34
2.84
6
0
0
100
2.99
7
0
0
166
3.1
8
0
0
170
3.18
9
0
0
110
3.23
10
0
0
44
3.28
11
0
0
10
3.31
12
0
0
1
3.33
Total
5
0
635
Other information about this multiloop Properties
Region degree sequence: [2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,1,2,3],[0,3,2,0],[0,1,4,5],[0,5,6,1],[2,6,7,8],[2,9,6,3],[3,5,9,4],[4,9,8,8],[4,7,7,9],[5,8,7,6]]
PD code (use to draw this multiloop with SnapPy): [[10,20,1,11],[11,9,12,10],[12,19,13,20],[1,8,2,9],[15,18,16,19],[13,7,14,8],[2,14,3,15],[17,5,18,6],[16,5,17,4],[6,3,7,4]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (10,11,-1,-12)(12,1,-13,-2)(18,3,-19,-4)(7,4,-8,-5)(16,5,-17,-6)(2,9,-3,-10)(8,19,-9,-20)(20,13,-11,-14)(17,14,-18,-15)(6,15,-7,-16)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,12)(-2,-10,-12)(-3,18,14,-11,10)(-4,7,15,-18)(-5,16,-7)(-6,-16)(-8,-20,-14,17,5)(-9,2,-13,20)(-15,6,-17)(-19,8,4)(1,11,13)(3,9,19)
Multiloop annotated with half-edges
12^2_78 annotated with half-edges